TWO PROPERTIES OF EXISTENTIALLY CLOSED COMPANIONS OF STRONGLY MINIMAL STRUCTURES

The proposed article studies some properties of existentially closed companions of strongly minimal structures. A criterion for the existential closedness of an arbitrary strongly minimal structure is found in the article and it is proved that the existentially closed companion of any strongly minimal structure is itself strongly minimal. It also follows from the resulting description that all existentially closed companions of a given strongly minimal structure form an axiomatizable class whose elementary theory is complete and model-complete and, therefore, coincides with its inductive and forcing companions. This is the reason for the importance of the work done and the high international significance of the theorems obtained in it. Another equally important consequence of this research is the discovery of an important subclass of strongly minimal theories. It should be noted that a complete description of this class of theories is an independent and extremely important task. It is known that natural numbers with the following relation are an example of a strongly minimal structure in which the existential type of zero is not minimal. Then the method used in the proof of the last theorem shows that the existentially closed companion of this structure are integers with the following relation.

On finite rigid structures

The main result of this paper is a probabilistic construction of finite rigid structures. It yields a finitely axiomatizable class of finite rigid structures where no formula with counting quantifiers defines a linear order.

Connections between axioms of set theory and basic theorems of universal algebra

One of the basic theorems in universal algebra is Birkhoff's variety theorem: the smallest equationally axiomatizable class containing a class K of algebras coincides with the class obtained by taking homomorphic images of subalgebras of direct products of elements of K. G. Grätzer asked whether the variety theorem is equivalent to the Axiom of Choice. In 1980, two of the present authors proved that Birkhoff's theorem can already be derived in ZF. Surprisingly, the Axiom of Foundation plays a crucial role here: we show that Birkhoff's theorem cannot be derived in ZF + AC \{Foundation}, even if we add Foundation for Finite Sets. We also prove that the variety theorem is equivalent to a purely set-theoretical statement, the Collection Principle. This principle is independent of ZF\{Foundation}. The second part of the paper deals with further connections between axioms of ZF-set theory and theorems of universal algebra.

An undecidable problem in finite combinatorics

Problems of computing probabilities of statements about large, finite structures have become an important subject area of finite combinatorics. Within the last two decades many researchers have turned their attention to such problems and have developed a variety of methods for dealing with them. Applications of these ideas include nonconstructive existence proofs for graphs with certain properties by showing the properties occur with nonzero probabilities (for examples see Erdös and Spencer [ES] and Bollobás [Bo]), and determination of average running times for sorting algorithms by computing asymptotic probabilities of statements about permutations (see Knuth [Kn]). Two types of techniques recur in solutions to such problems: probabilistic techniques, such as those used by Erdös and Spencer [ES], and classical assymptotic techniques, such as those surveyed by Bender [Be] and de Bruijn [Br]. Studying this body of techniques, one notices that characteristics of these problems suggest certain methods of solution, in much the same way that the form of an integrand may suggest certain substitutions. The question arises, then, as to whether there is a systematic way to approach these problems: is there an algorithm for computing asymptotic probabilities? I will show that the answer is “no”—for any reasonable formulation, the problem of computing asymptotic probabilities is undecidable.The main theorem of the paper is Theorem 1.6, which says that there is a finitely axiomatizable class in which every first order sentence has an asymptotic probability of 0 or 1—i.e., is almost always true or almost always false in finite structures—but for which the problem of deciding whether a sentence has asymptotic probability 0 or 1 is undecidable. Heretofore, classes known to have such a 0-1 law have had decidable asymptotic probability problems (see Lynch [Ly] for examples and a discussion of previous work in the area).